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Multivariable Calculus/Calculus & Geometry textbook suggestions

  1. May 17, 2011 #1
    Hello all,

    I am currently in University and I am retaking a course that I dropped last year due to the fact that there was no textbook and the time I spent in the course was a frantic scramble for reliable information and attempting to understand what the professor was teaching. I have looked before and the only textbook that I found that was remotely close to the material that is covered in the course is Multivariable Calculus by Edwards and Penney. I am asking the community for any suggestions for textbooks that cover most, if not all of the topics listed below:

    - Differentiation in Several Variables
    Partial derivatives, directional derivatives, del operator, Mean-value theorem for a function of several variables, differentiation through an integral, Leibniz's Rule 2nd order derivatives and Clairaut's theorem, Hessian Matrix. Functions from several variables to several variables, Jacobian Matrix

    - Inverse - and Implicit functions in several variables
    Inverse function theorem
    Implicit function theorem

    - Quadratic forms
    Matrix Representation
    Change of variable & diagonalization by congruence
    Positive & Negative definiteness, Semi-definieness, determinantal criteria
    Sylvester's Theorem

    - Extrema
    Local extrema-critical points, local max, local min, saddle points, determinantal tests on the Hessian Matrix
    Global extrema-existence for continuous functwions on closed, bounded sets, finiding constrained extrema by the method of Lagrange multipliers

    - Curves and Surfaces
    Space curves defined parametrically-tangent, normal, binormal, arc length, curvature, torsion
    Surfaces-implicit and parametric definitions-tangent plane, normal line
    Space curves defined as intersecting surgaces, related quadratic & cubic forms

    - Integration in several Variables
    Multiple and iterated integrals
    Change of order of variables
    Change of variable formula in general
    Polar, cylindrical & spherical coordinates
    Line integrals, consecutive and non-consecutive vector fields, curl operator
    Surface integrals, projection onto a plane, parametric surgace integrals
    Stokes' theorem, boundary curve, orientation
    Gauss' theorem, boundary surgace, div operator

    Any suggestions will be greatly appreciated and sorry for the long list, I didn't want to leave anything out.
  2. jcsd
  3. May 17, 2011 #2


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  4. May 18, 2011 #3
    I fully second Apostol and Courant.
  5. May 18, 2011 #4
    Apostol and Courant are excellent but they are mathematically rigorous and I don't know if that is what you want. If not, then there is the book by Schey Div, Grad, Curl and All That for intuition; Marsden and Tromba Vector Calculus is an intermediate sort of book; and the old favourite is Stewart Multivariable Calculus or the relevant chapters in his Calculus. Stewart gives the simplest presentation of calculus.

    I suggest going to the library and looking through all these books and deciding for yourself which one best suites your needs.
  6. May 18, 2011 #5
    any multivariable calculus text will fit your description..
  7. Jun 28, 2012 #6
    I agree that Courant and Apostol are NOT light reads. Stewart has an excellent MV book that is good for all audiences. A lesser known book is Howard Anton's Calculus, A new horizon, which has a lot of good applications of calc added in. Of course the Khan Academy has very easy-to-understand lectures, but not in all of the above mentioned topics.
  8. Jul 1, 2012 #7
  9. Jul 1, 2012 #8


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  10. Jul 2, 2012 #9
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